Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

Unsorted Functional Translations

AbstractIn this article we first show how the functional and the optimized functional translation from modal logic to many-sorted first-order logic can be naturally extended to the hybrid language H(@,↓). The translation is correct not only when reasoning over the class of all models, but for any first-order definable class. We then show that sorts can be safely removed (i.e., without affecting the satisfiability status of the formula) for frame classes that can be defined in the basic modal language, and show a counterexample for a frame class defined using nominals

Synthesizing semantics for extensions of propositional logic

Given a Hilbert style specification of a propositional extension of standard propositional logic, it is shown how the basic model theoretic semantics can be obtained from the axioms by syntactic transformations. The transformations are designed in such a way that they eliminate certain derived theorems from the Hilbert axiomatization by turning them into tautologies. The following transformations are considered. Elimination of the reflexivity and transitivity of a binary consequence relation yields the basic possible worlds framework. Elimination of the congruence properties of the connectives yields weak neighbourhood semantics. Elimination of certain monotonicity properties yields a stronger neighbourhood semantics. Elimination of certain closure properties yields relational possible worlds semantics for the connectives. If propositional logic is the basis of the specification, the translated Hilbert axioms can be simplified by eliminating the formula variables with a quantifier elimination algorithm. This way we obtain the frame conditions for the semantic structures. All transformations work for arbitrary n-place connectives. The steps can be fully automated by means of PL1 theorem provers and quantifier elimination algorithms. The meta theory guarantees soundness and completeness of all transformation steps. As a by--product, translations into multi--modal logic are developed

Automated Reasoning in Deontic Logic

Deontic logic is a very well researched branch of mathematical logic and philosophy. Various kinds of deontic logics are discussed for different application domains like argumentation theory, legal reasoning, and acts in multi-agent systems. In this paper, we show how standard deontic logic can be stepwise transformed into description logic and DL- clauses, such that it can be processed by Hyper, a high performance theorem prover which uses a hypertableau calculus. Two use cases, one from multi-agent research and one from the development of normative system are investigated

A Two-Level Logic Approach to Reasoning about Typed Specification Languages

International audienceThe two-level logic approach (2LL) to reasoning about computational specifications, as implemented by the Abella theorem prover, represents derivations of a specification language as an inductive definition in a reasoning logic. This approach has traditionally been formulated with the specification and reasoning logics having the same type system, and only the formulas being translated. However, requiring identical type systems limits the approach in two important ways: (1) every change in the specification language's type system requires a corresponding change in that of the reasoning logic, and (2) the same reasoning logic cannot be used with two specification languages at once if they have incompatible type systems. We propose a technique based on adequate encodings of the types and judgements of a typed specification language in terms of a simply typed higher-order logic program, which is then used for reasoning about the specification language in the usual 2LL. Moreover, a single specification logic implementation can be used as a basis for a number of other specification languages just by varying the encoding. We illustrate our technique with an implementation of the LF dependent type theory as a new specification language for Abella, co-existing with its current simply typed higher-order hereditary Harrop specification logic, without modifying the type system of its reasoning logic